Causal Universes

Fol­lowup to: Stuff that Makes Stuff Happen

Pre­vi­ous med­i­ta­tion: Does the idea that ev­ery­thing is made of causes and effects mean­ingfully con­strain ex­pe­rience? Can you co­her­ently say how re­al­ity might look, if our uni­verse did not have the kind of struc­ture that ap­pears in a causal model?

I can de­scribe to you at least one fa­mous uni­verse that didn’t look like it had causal struc­ture, namely the uni­verse of J. K. Rowl­ing’s Harry Pot­ter.

You might think that J. K. Rowl­ing’s uni­verse doesn’t have causal struc­ture be­cause it con­tains magic—that wiz­ards wave their wands and cast spells, which doesn’t make any sense and goes against all sci­ence, so J. K. Rowl­ing’s uni­verse isn’t ‘causal’.

In this you would be com­pletely mis­taken. The do­main of “causal­ity” is just “stuff that makes stuff hap­pen and hap­pens be­cause of other stuff”. If Dum­ble­dore waves his wand and there­fore a rock floats into the air, that’s causal­ity. You don’t even have to use words like ‘there­fore’, let alone big fancy phrases like ‘causal pro­cess’, to put some­thing into the lofty-sound­ing do­main of causal­ity. There’s causal­ity any­where there’s a noun, a verb, and a sub­ject: ‘Dum­ble­dore’s wand lifted the rock.’ So far as I could tell, there wasn’t any­thing in Lord of the Rings that vi­o­lated causal­ity.

You might worry that J. K. Rowl­ing had made a con­ti­nu­ity er­ror, de­scribing a spell work­ing one way in one book, and a differ­ent way in a differ­ent book. But we could just sup­pose that the spell had changed over time. If we ac­tu­ally found our­selves in that ap­par­ent uni­verse, and saw a spell have two differ­ent effects on two differ­ent oc­ca­sions, we would not con­clude that our uni­verse was un­com­putable, or that it couldn’t be made of causes and effects.

No, the only part of J. K. Rowl­ing’s uni­verse that vi­o­lates ‘cause and effect’ is...

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...the Time-Turn­ers, of course.

A Time-Turner, in Rowl­ing’s uni­verse, is a small hour­glass neck­lace that sends you back in time 1 hour each time you spin it. In Rowl­ing’s uni­verse, this time-travel doesn’t al­low for chang­ing his­tory; what­ever you do af­ter you go back, it’s already hap­pened. The uni­verse con­tain­ing the time-travel is a sta­ble, self-con­sis­tent ob­ject.

If a time ma­chine does al­low for chang­ing his­tory, it’s easy to imag­ine how to com­pute it; you could eas­ily write a com­puter pro­gram which would simu­late that uni­verse and its time travel, given suffi­cient com­put­ing power. You would store the state of the uni­verse in RAM and simu­late it un­der the pro­grammed ‘laws of physics’. Every nanosec­ond, say, you’d save a copy of the uni­verse’s state to disk. When the Time-Changer was ac­ti­vated at 9pm, you’d re­trieve the saved state of the uni­verse from one hour ago at 8pm, load it into RAM, and then in­sert the Time-Changer and its user in the ap­pro­pri­ate place. This would, of course, dump the rest of the uni­verse from 9pm into oblivion—no pro­cess­ing would con­tinue on­ward from that point, which is the same as end­ing that world and kil­ling ev­ery­one in it.[1]

Still, if we don’t worry about the ethics or the disk space re­quire­ments, then a Time-Changer which can re­store and then change the past is easy to com­pute. There’s a perfectly clear or­der of causal­ity in meta­time, in the lin­ear time of the simu­lat­ing com­puter, even if there are ap­par­ent cy­cles as seen from within the uni­verse. The per­son who sud­denly ap­pears with a Time-Changer is the causal de­scen­dant of the older uni­verse that just got dumped from RAM.

But what if in­stead, re­al­ity is always—some­how—perfectly self-con­sis­tent, so that there’s ap­par­ently only one uni­verse with a fu­ture and a past that never changes, so that the per­son who ap­pears at 8PM has always seem­ingly de­scended from the very same uni­verse that then de­vel­ops by 9PM...?

How would you com­pute that in one sweep-through, with­out any higher-or­der meta­time?

What would a causal graph for that look like, when the past de­scends from its very own fu­ture?

And the an­swer is that there isn’t any such causal graph. Causal mod­els are some­times referred to as DAGs, which stands for Directed Acyclic Graph. If in­stead there’s a di­rected cy­cle, there’s no ob­vi­ous or­der in which to com­pute the joint prob­a­bil­ity table. Even if you some­how knew that at 8PM some­body was go­ing to ap­pear with a Time-Turner used at 9PM, you still couldn’t com­pute the ex­act state of the time-trav­el­ler with­out already know­ing the fu­ture at 9PM, and you couldn’t com­pute the fu­ture with­out know­ing the state at 8PM, and you couldn’t com­pute the state at 8PM with­out know­ing the state of the time-trav­el­ler who just ar­rived.

In a causal model, you can com­pute p(9pm|8pm) and p(8pm|7pm) and it all starts with your un­con­di­tional knowl­edge of p(7pm) or per­haps the Big Bang, but with a Time-Turner we have p(9pm|8pm) and p(8pm|9pm) and we can’t un­tan­gle them—mul­ti­ply­ing those two con­di­tional ma­tri­ces to­gether would just yield non­sense.

Does this mean that the Time-Turner is be­yond all logic and rea­son?

Com­plete philo­soph­i­cal panic is ba­si­cally never jus­tified. We should even be re­luc­tant to say any­thing like, “The so-called Time-Turner is be­yond co­her­ent de­scrip­tion; we only think we can imag­ine it, but re­ally we’re just talk­ing non­sense; so we can con­clude a pri­ori that no such Time-Turner that can ex­ist; in fact, there isn’t even a mean­ingful thing that we’ve just proven can’t ex­ist.” This is also panic—it’s just been made to sound more dig­nified. The first rule of sci­ence is to ac­cept your ex­per­i­men­tal re­sults, and gen­er­al­ize based on what you see. What if we ac­tu­ally did find a Time-Turner that seemed to work like that? We’d just have to ac­cept that Causal­ity As We Pre­vi­ously Knew It had gone out the win­dow, and try to make the best of that.

In fact, de­spite the some­what-jus­tified con­cep­tual panic which the pro­tag­o­nist of Harry Pot­ter and the Meth­ods of Ra­tion­al­ity un­der­goes upon see­ing a Time-Turner, a uni­verse like that can have a straight­for­ward log­i­cal de­scrip­tion even if it has no causal de­scrip­tion.


Con­way’s Game of Life is a very sim­ple speci­fi­ca­tion of a causal uni­verse; what we would to­day call a cel­lu­lar au­toma­ton. The Game of Life takes place on a two-di­men­sional square grid, so that each cell is sur­rounded by eight oth­ers, and the Laws of Physics are as fol­lows:

  • A cell with 2 liv­ing neigh­bors dur­ing the last tick, re­tains its state from the last tick.

  • A cell with 3 liv­ing neigh­bors dur­ing the last tick, will be al­ive dur­ing the next tick.

  • A cell with fewer than 2 or more than 3 liv­ing neigh­bors dur­ing the last tick, will be dead dur­ing the next tick.

It is my con­sid­ered opinion that ev­ery­one should play around with Con­way’s Game of Life at some point in their lives, in or­der to com­pre­hend the no­tion of ‘laws of physics’. Play­ing around with Life as a kid (on a Mac Plus) helped me gut-level-un­der­stand the con­cept of a ‘lawful uni­verse’ de­vel­op­ing un­der ex­cep­tion­less rules.

Now sup­pose we mod­ify the Game of Life uni­verse by adding some pre­speci­fied cases of time travel—places where a cell will de­scend from neigh­bors in the fu­ture, in­stead of the past.

In par­tic­u­lar we shall take a 4x4 Life grid, and ar­bi­trar­ily hack Con­way’s rules to say:

  • On the 2nd tick, the cell at (2,2) will have its state de­ter­mined by that cell’s state on the 3rd tick, in­stead of its neigh­bors on the 1st tick.

It’s no longer pos­si­ble to com­pute the state of each cell at each time in a causal or­der where we start from known cells and com­pute their not-yet-known causal de­scen­dants. The state of the cells on the 3rd tick, de­pend on the state of the cells on the 2nd tick, which de­pends on the state on the 3rd tick.

In fact, the time-travel rule, on the same ini­tial con­di­tions, also per­mits a live cell to travel back in time, not just a dead cell—this just gives us the “nor­mal” grid! Since you can’t com­pute things in or­der of cause and effect, even though each lo­cal rule is de­ter­minis­tic, the global out­come is not de­ter­mined.

How­ever, you could simu­late Life with time travel merely by brute-force search­ing through all pos­si­ble Life-his­to­ries, dis­card­ing all his­to­ries which di­s­obeyed the laws of Life + time travel. If the en­tire uni­verse were a 4-by-4 grid, it would take 16 bits to spec­ify a sin­gle slice through Time—the uni­verse’s state dur­ing a sin­gle clock tick. If the whole of Time was only 3 ticks long, there would be only 48 bits mak­ing up a can­di­date ‘his­tory of the uni­verse’ - it would only take 48 bits to com­pletely spec­ify a His­tory of Time. 2^48 is just 281,474,976,710,656, so with a cluster of 2GHz CPUs it would be quite prac­ti­cal to find, for this rather tiny uni­verse, the set of all pos­si­ble his­to­ries that obey the log­i­cal re­la­tions of time travel.

It would no longer be pos­si­ble to point to a par­tic­u­lar cell in a par­tic­u­lar his­tory and say, “This is why it has the ‘al­ive’ state on tick 3”. There’s no “rea­son”—in the frame­work of causal rea­sons—why the time-trav­el­ing cell is ‘dead’ rather than ‘al­ive’, in the his­tory we showed. (Well, ex­cept that Alex, in the real uni­verse, hap­pened to pick it out when I asked him to gen­er­ate an ex­am­ple.) But you could, in prin­ci­ple, find out what the set of per­mit­ted his­to­ries for a large digi­tal uni­verse, given lots and lots of com­put­ing power.

Here’s an in­ter­est­ing ques­tion I do not know how to an­swer: Sup­pose we had a more com­pli­cated set of cel­lu­lar au­toma­ton rules, on a vastly larger grid, such that the cel­lu­lar au­toma­ton was large enough, and sup­ported enough com­plex­ity, to per­mit peo­ple to ex­ist in­side it and be com­puted. Pre­sum­ably, if we com­puted out cell states in the or­di­nary way, each fu­ture fol­low­ing from its im­me­di­ate past, the peo­ple in­side it would be as real as we hu­mans com­puted un­der our own uni­verse’s causal physics.

Now sup­pose that in­stead of com­put­ing the cel­lu­lar au­toma­ton causally, we hack the rules of the au­toma­ton to add large time-travel loops—change their physics to al­low Time-Turn­ers—and with an un­rea­son­ably large com­puter, the size of two to the power of the num­ber of bits com­pris­ing an en­tire his­tory of the cel­lu­lar au­toma­ton, we enu­mer­ate all pos­si­ble can­di­dates for a uni­verse-his­tory.

So far, we’ve just gen­er­ated all 2^N pos­si­ble bit­strings of size N, for some large N; noth­ing more. You wouldn’t ex­pect this pro­ce­dure to gen­er­ate any peo­ple or make any ex­pe­riences real, un­less enu­mer­at­ing all finite strings of size N causes all lawless uni­verses en­coded in them to be real. There’s no causal­ity there, no com­pu­ta­tion, no law re­lat­ing one time-slice of a uni­verse to the next...

Now we set the com­puter to look over this en­tire set of can­di­dates, and mark with a 1 those that obey the mod­ified re­la­tions of the time-trav­el­ing cel­lu­lar au­toma­ton, and mark with a 0 those that don’t.

If N is large enough—if the size of the pos­si­ble uni­verse and its du­ra­tion is large enough—there would be de­scrip­tions of uni­verses which ex­pe­rienced nat­u­ral se­lec­tion, evolu­tion, per­haps the evolu­tion of in­tel­li­gence, and of course, time travel with self-con­sis­tent Time-Turn­ers, obey­ing the mod­ified re­la­tions of the cel­lu­lar au­toma­ton. And the checker would mark those de­scrip­tions with a 1, and all oth­ers with a 0.

Sup­pose we pick out one of the his­to­ries marked with a 1 and look at it. It seems to con­tain a de­scrip­tion of peo­ple who re­mem­ber ex­pe­rienc­ing time travel.

Now, were their ex­pe­riences real? Did we make them real by mark­ing them with a 1 - by ap­ply­ing the log­i­cal filter us­ing a causal com­puter? Even though there was no way of com­put­ing fu­ture events from past events; even though their uni­verse isn’t a causal uni­verse; even though they will have had ex­pe­riences that liter­ally were not ‘caused’, that did not have any causal graph be­hind them, within the frame­work of their own uni­verse and its rules?


I don’t know. But...

Our own uni­verse does not ap­pear to have Time-Turn­ers, and does ap­pear to have strictly lo­cal causal­ity in which each vari­able can be com­puted strictly for­ward-in-time.

And I don’t know why that’s the case; but it’s a likely-look­ing hint for any­one won­der­ing what sort of uni­verses can be real in the first place.

The col­lec­tion of hy­po­thet­i­cal math­e­mat­i­cal thin­gies that can be de­scribed log­i­cally (in terms of re­la­tional rules with con­sis­tent solu­tions) looks vastly larger than the col­lec­tion of causal uni­verses with lo­cally de­ter­mined, acycli­cally or­dered events. Most math­e­mat­i­cal ob­jects aren’t like that. When you say, “We live in a causal uni­verse”, a uni­verse that can be com­puted in-or­der us­ing lo­cal and di­rec­tional rules of de­ter­mi­na­tion, you’re vastly nar­row­ing down the pos­si­bil­ities rel­a­tive to all of Math-space.

So it’s rather sug­ges­tive that we find our­selves in a causal uni­verse rather than a log­i­cal uni­verse—it sug­gests that not all math­e­mat­i­cal ob­jects can be real, and the sort of thin­gies that can be real and have peo­ple in them are con­strained to some­where in the vicinity of ‘causal uni­verses’. That you can’t have con­scious­ness with­out com­put­ing an agent made of causes and effects, or maybe some­thing can’t be real at all un­less it’s a fabric of cause and effect. It sug­gests that if there is a Teg­mark Level IV mul­ti­verse, it isn’t “all log­i­cal uni­verses” but “all causal uni­verses”.

Of course you also have to be a bit care­ful when you start as­sum­ing things like “Only causal things can be real” be­cause it’s so easy for Real­ity to come back at you and shout “WRONG!” Sup­pose you thought re­al­ity had to be a dis­crete causal graph, with a finite num­ber of nodes and dis­crete de­scen­dants, ex­actly like Pearl-stan­dard causal mod­els. There would be no hy­poth­e­sis in your hy­poth­e­sis-space to de­scribe the stan­dard model of physics, where space is con­tin­u­ous, in­definitely di­visi­ble, and has com­plex am­pli­tude as­sign­ments over un­countable car­di­nal­ities of points.

Real­ity is pri­mary, saith the wise old mas­ters of sci­ence. The first rule of sci­ence is just to go with what you see, and try to un­der­stand it; rather than stand­ing on your as­sump­tions, and try­ing to ar­gue with re­al­ity.

But even so, it’s in­ter­est­ing that the pure, ideal struc­ture of causal mod­els, in­vented by statis­ti­ci­ans to reify the idea of ‘causal­ity’ as sim­ply as pos­si­ble, looks much more like the mod­ern view of physics than does the old New­to­nian ideal.

If you be­lieved in New­to­nian billiard balls bounc­ing around, and some­body asked you what sort of things can be real, you’d prob­a­bly start talk­ing about ‘ob­jects’, like the billiard balls, and ‘prop­er­ties’ of the ob­jects, like their lo­ca­tion and ve­loc­ity, and how the lo­ca­tion ‘changes’ be­tween one ‘time’ and an­other, and so on.

But sup­pose you’d never heard of atoms or ve­loc­i­ties or this ‘time’ stuff—just the causal di­a­grams and causal mod­els in­vented by statis­ti­ci­ans to rep­re­sent the sim­plest pos­si­ble cases of cause and effect. Like this:

And then some­one says to you, “In­vent a con­tin­u­ous analogue of this.”

You wouldn’t in­vent billiard balls. There’s no billiard balls in a causal di­a­gram.

You wouldn’t in­vent a sin­gle time sweep­ing through the uni­verse. There’s no sweep­ing time in a causal di­a­gram.

You’d stare a bit at B, C, and D which are the sole nodes de­ter­min­ing A, screen­ing off the rest of the graph, and say to your­self:

“Okay, how can I in­vent a con­tin­u­ous analogue of there be­ing three nodes that screen off the rest of the graph? How do I do that with a con­tin­u­ous neigh­bor­hood of points, in­stead of three nodes?”

You’d stare at E de­ter­min­ing D de­ter­min­ing A, and ask your­self:

“How can I in­vent a con­tin­u­ous analogue of ‘de­ter­mi­na­tion’, so that in­stead of E de­ter­min­ing D de­ter­min­in­ing A, there’s a con­tinuum of de­ter­mined points be­tween E and A?”

If you gen­er­al­ized in a cer­tain sim­ple and ob­vi­ous fash­ion...

The con­tinuum of re­lat­ed­ness from B to C to D would be what we call space.

The con­tinuum of de­ter­mi­na­tion from E to D to A would be what we call time.

There would be a rule stat­ing that for ep­silon time be­fore A, there’s a neigh­bor­hood of spa­tial points delta which screens off the rest of the uni­verse from be­ing rele­vant to A (so long as no de­scen­dants of A are ob­served); and that ep­silon and delta can both get ar­bi­trar­ily close to zero.

There might be—if you were just pick­ing the sim­plest rules you could man­age—a phys­i­cal con­stant which re­lated the met­ric of re­lat­ed­ness (space) to the met­ric of de­ter­mi­na­tion (time) and so en­forced a sim­ple con­tin­u­ous analogue of lo­cal causal­ity...

...in our uni­verse, we call it c, the speed of light.

And it’s worth re­mem­ber­ing that Isaac New­ton did not ex­pect that rule to be there.

If we just stuck with Spe­cial Rel­a­tivity, and didn’t get any more mod­ern than that, there would still be lit­tle billiard balls like elec­trons, oc­cu­py­ing some par­tic­u­lar point in that neigh­bor­hood of space.

But if your lit­tle neigh­bor­hoods of space have billiard balls with ve­loc­i­ties, many of which are slower than light­speed… well, that doesn’t look like the sim­plest con­tin­u­ous analogues of a causal di­a­gram, does it?

When we make the first quan­tum leap and de­scribe par­ti­cles as waves, we find that the billiard balls have been elimi­nated. There’s no ‘par­ti­cles’ with a sin­gle point po­si­tion and a ve­loc­ity slower than light. There’s an elec­tron field, and waves prop­a­gate through the elec­tron field through points in­ter­act­ing only with lo­cally neigh­bor­ing points. If a par­tic­u­lar elec­tron seems to be mov­ing slower than light, that’s just be­cause—even though causal­ity always prop­a­gates at ex­actly c be­tween points within the elec­tron field—the crest of the elec­tron wave can ap­pear to move slower than that. A billiard ball mov­ing through space over time, has been re­placed by a set of points with val­ues de­ter­mined by their im­me­di­ate his­tor­i­cal neigh­bor­hood.

vs.

And when we make the sec­ond quan­tum leap into con­figu­ra­tion space, we find a time­less uni­ver­sal wave­func­tion with com­plex am­pli­tudes as­signed over the points in that con­figu­ra­tion space, and the am­pli­tude of ev­ery point causally de­ter­mined by its im­me­di­ate neigh­bor­hood in the con­figu­ra­tion space.[2]

So, yes, Real­ity can poke you in the nose if you de­cide that only dis­crete causal graphs can be real, or some­thing silly like that.

But on the other hand, tak­ing ad­vice from the math of causal­ity wouldn’t always lead you astray. Modern physics looks a heck of a lot more similar to “Let’s build a con­tin­u­ous analogue of the sim­plest di­a­grams statis­ti­ci­ans in­vented to de­scribe the­o­ret­i­cal causal­ity”, than like any­thing New­ton or Aris­to­tle imag­ined by look­ing at the ap­par­ent world of boulders and planets.

I don’t know what it means… but per­haps we shouldn’t ig­nore the hint we re­ceived by virtue of find­ing our­selves in­side the nar­row space of “causal uni­verses”—rather than the much wider space “all log­i­cal uni­verses”—when it comes to guess­ing what sort of thin­gies can be real. To the ex­tent we al­low non-causal uni­verses in our hy­poth­e­sis space, there’s a strong chance that we are broad­en­ing our imag­i­na­tion be­yond what can re­ally be real un­der the Ac­tual Rules—what­ever they are! (It is pos­si­ble to broaden your meta­physics too much, as well as too lit­tle. For ex­am­ple, you could al­low log­i­cal con­tra­dic­tions into your hy­poth­e­sis space—col­lec­tions of ax­ioms with no mod­els—and ask whether we lived in one of those.)

If we trusted ab­solutely that only causal uni­verses could be real, then it would be safe to al­low only causal uni­verses into our hy­poth­e­sis space, and as­sign prob­a­bil­ity liter­ally zero to ev­ery­thing else.

But if you were scared of be­ing wrong, then as­sign­ing prob­a­bil­ity liter­ally zero means you can’t change your mind, ever, even if Pro­fes­sor McGon­a­gall shows up with a Time-Turner to­mor­row.

Med­i­ta­tion: Sup­pose you needed to as­sign non-zero prob­a­bil­ity to any way things could con­ceiv­ably turn out to be, given hu­man­ity’s rather young and con­fused state—enu­mer­ate all the hy­pothe­ses a su­per­in­tel­li­gent AI should ever be able to ar­rive at, based on any sort of strange world it might find by ob­ser­va­tion of Time-Turn­ers or stranger things. How would you enu­mer­ate the hy­poth­e­sis space of all the wor­lds we could re­motely maybe pos­si­bly be liv­ing in, in­clud­ing wor­lds with hy­per­com­put­ers and Stable Time Loops and even stranger fea­tures?

Main­stream sta­tus.


[1] Some­times I still mar­vel about how in most time-travel sto­ries no­body thinks of this. I guess it re­ally is true that only peo­ple who are sen­si­tized to ‘think­ing about ex­is­ten­tial risk’ even no­tice when a world ends, or when billions of peo­ple are ex­tin­guished and re­placed by slightly differ­ent ver­sions of them­selves. But then al­most no­body will no­tice that sort of thing in­side their fic­tion if the char­ac­ters all act like it’s okay.)

[2] Un­less you be­lieve in ‘col­lapse’ in­ter­pre­ta­tions of quan­tum me­chan­ics where Bell’s The­o­rem math­e­mat­i­cally re­quires that ei­ther your causal mod­els don’t obey the Markov con­di­tion or they have faster-than-light non­lo­cal in­fluences. (De­spite a large liter­a­ture of ob­scu­ran­tist ver­bal words in­tended to ob­scure this fact, as gen­er­ated and con­sumed by physi­cists who don’t know about for­mal defi­ni­tions of causal­ity or the Markov con­di­tion.) If you be­lieve in a col­lapse pos­tu­late, this whole post goes out the win­dow. But frankly, if you be­lieve that, you are bad and you should feel bad.

Part of the se­quence Highly Ad­vanced Episte­mol­ogy 101 for Beginners

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